434 HlSTOBY OF THE THEORY OF NuMBEKS. [CHAP. XVIII Hence the number of integers £x which are divisible by a, but not by a — 1, a— 2,. .., 2, is [*/a] o-l aa=s n (!-«„). >(»), of primes ^x and >? = [Vz] is M — 1— SIIJ0-.. Let 'ie primes ^ Va« Let P« be the greatest prime g v. Then fan " t'&'^L ,rwfc~n -i-M- « -s 2 nn-A ^- [. L^J o»3Ai=l /i=2L LPMJJ jegendre's formula. otained a formula to compute the number of primes >resupposing a knowledge of any primes >2, by consid->ositive integers n, n', . . . for which yd with Legendre's formula for the number A (z) of d the remainders t—\t\, and wrote Rn(z) for the sum inders. He obtained relations between values of the is arguments z, and treated sums of such values. For •OAO), Pn-fl-1 S' (xv+1 - xv)A (v) = -2xp+xPn+1 A (pn) , l"»l summed for the primes p between 1 and the nth prime pn. By special choice of the z's, we get formulas involving Euler's ^-function (p. 1818), and the number or sum of the divisors of an integer. See Rogel22 of Ch. XI. G. Andreoli244 noted that, if x is real, and T is the gamma function, X is zero if and only if x is a prime. Hence the number of primes <n is The sum of the fcth powers of the primes <n is given asymptotically. M. Petrovitch245 used a real function 6(x, u), like a cos 2irx-\-b cos 2iru—a — b, which is zero for every pair of integers x, u, and not if x or u is fractional. Let$(x) be the function obtained from 6(x, u) by taking u={l+T(x)}/x. Thus 2/=$(x) cuts the re-axis in points whose abscissas are the primes. '"Giornale di Mat., 47, 1909, 305-320. M'Sitzungsber. Ak. Wiss. Wien (Math.), 121, 1912, II a, 1785-1824; 122, 1913, II a, 669-700. 2«Rendiconti Accad. Lincei, (5), 21, II, 1912, 404-7. Wigert.236a "BNouv. Ann. Math., (4), 13, 1913, 406-10.